Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C7×#340 ⋊ C2 |
| Sylow | C7 ⋊ #1999 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | D4×D7 |
| commutator, cosocle | C22×C4×C7 | C23 |
| Frattini | C22×C4 | C22×D7 |
| Fitting | C7×#340 | C2 |
| socle | C23×C7 | C2×D4 |
| hypercenter | #340 | D7 |
| hypocenter | C7 | #1999 |
| LCS | G ▷ C22×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C22×C4 ◁ #340 | |
496 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 448 |
Polycyclic presentation:
2^7,7|d^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=h^6,c^b=cf,d^b=dg,h^b=h^6,h^c=h^6116 conjugacy classes — view the full character table.
| maximal subgroup of | #13265894, #13265899, #13265905, … |
| maximal normal subgroup of | #13265894, #13265899, #13265905, … |
| maximal quotient of | #13253995, #13254002, #13254023, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 1406);